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Is Math flawed at the extremes?

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sravna

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Mathematics can be used to model reality. In my view there is a fundamental problem in Math that does not allow it to be a good model of reality especially when it comes to describing the inordinately that is in mathematical terms infinitely high or low values.

The question is can anything be infinitely large, long etc. or infinitesimally small, short etc.? Let us take the example of the structure of space. We know a space is characterized by distance between any two points. Now try to think of how small this distance can be. Logically at a certain point it has to be zero.

But if you mathematically approach it, the values can go on indefinitely small. For example it can start from 1 then go to 0.5 to 0.25 to .125 and so on. So when does the distance become 0? The solution I think is to assume that the above representation through Math does not work at the lowest level and at some point, i.e, at some threshold the distance becomes suddenly 0. That is it the space abdicates its nature of presenting itself as distance and a higher nature of space reveals itself.

Similarly consider the speed of light. It is the fastest a physical energy can travel. So nothing can travel at infinite speed just as we saw above that it is not possible to be indefinitely small. Infinite speed would not make sense. That means you can travel faster than you always did. Where would the energy come from?. In a way similar to the above the infinity does not reflect the reality. I think there is evidence that thoughts are instantly everywhere. I cannot vouch for the veracity of this but it seems to me to be reasonable as a truth. So in this case physical energy transcending into a higher mental energy and neatly solving the problem of infinite speed. At extremes these interesting things seem to be happening.

So I would conclude that any theory that makes use of the concept such as infinity will not lead to a real description of the reality and we seriously need to assess alternatives to model reality in very extreme cases.
 
Mathematics can be used to model reality. In my view there is a fundamental problem in Math that does not allow it to be a good model of reality especially when it comes to describing the inordinately that is in mathematical terms infinitely high or low values.

Dear sravna,

From a layman's point I give my views (for a moment I keep away all that I have accumulated as scientific knowledge) though I may be using scientific terms because of habit and for convenience.

The question is can anything be infinitely large, long etc. or infinitesimally small, short etc.? Let us take the example of the structure of space. We know a space is characterized by distance between any two points. Now try to think of how small this distance can be. Logically at a certain point it has to be zero.

It will be zero when one point you have taken superimposes itself on the other or when one point overlaps on the other point. Without any of this happening there will always be space between the two points however it will be infinitesimally small. If you observe with a human eye the difference may not be visible at very small distances because the resolution power of the lens in your eyes is limited. May be a tiger moth can see much shorter distances with its revolving eyes which have multiple elements, each element a separate eye wired to its "brain". Alternatively you can view this under a microscope with a large resolution power.


But if you mathematically approach it, the values can go on indefinitely small. For example it can start from 1 then go to 0.5 to 0.25 to .125 and so on. So when does the distance become 0? The solution I think is to assume that the above representation through Math does not work at the lowest level and at some point, i.e, at some threshold the distance becomes suddenly 0. That is it the space abdicates its nature of presenting itself as distance and a higher nature of space reveals itself.

I do not think it will suddenly become zero as you have suggested. I believe it will still be incomprehensibly small for a human mind. With some effort it should be possible to represent this fact mathematically and build models to deal with situations at that micro level. I think people have done that.

Similarly consider the speed of light. It is the fastest a physical energy can travel. So nothing can travel at infinite speed just as we saw above that it is not possible to be indefinitely small. Infinite speed would not make sense. That means you can travel faster than you always did. Where would the energy come from?. In a way similar to the above the infinity does not reflect the reality. I think there is evidence that thoughts are instantly everywhere. I cannot vouch for the veracity of this but it seems to me to be reasonable as a truth. So in this case physical energy transcending into a higher mental energy and neatly solving the problem of infinite speed. At extremes these interesting things seem to be happening.

You speak about speed. Speed is relative and not absolute. Even leaving that, if you have infinite space you can have infinite speed too. A particle can start from rest and keep on accelerating towards reaching that point at the infinite distance in the infinite space. You have to only wait till infinity for the particle to reach that infinite speed because it will still be accelerating towards infinity. I am going mad. I am just a human being. I have limits.

So I would conclude that any theory that makes use of the concept such as infinity will not lead to a real description of the reality and we seriously need to assess alternatives to model reality in very extreme cases.

I take a paper and keeping it on a table top I draw a line on it with a scale. I make a mark of two points 2 inches apart and connect them by a straight line with the scale. I tell the student that the shortest distance between the points is a straight line and move on to explain more complicated concepts starting with that simple beginning. But I also know that my shortest distance is not really shortest consdering nthe fact that it is just an arc (an infinitesimal one) of a huge circle which represents the surface of the earth. But it is convenient to work with that axiom. So when we really come to deal with the expanding universe we will have to give more attention to the problem of measurement of distances. For the present we are at the beginning of the understanding-about the expanding universe.

Some loud thinking by a lay man.Thanks.
 
Dear sravna,

From a layman's point I give my views (for a moment I keep away all that I have accumulated as scientific knowledge) though I may be using scientific terms because of habit and for convenience.



It will be zero when one point you have taken superimposes itself on the other or when one point overlaps on the other point. Without any of this happening there will always be space between the two points however it will be infinitesimally small. If you observe with a human eye the difference may not be visible at very small distances because the resolution power of the lens in your eyes is limited. May be a tiger moth can see much shorter distances with its revolving eyes which have multiple elements, each element a separate eye wired to its "brain". Alternatively you can view this under a microscope with a large resolution power.

I do not think it will suddenly become zero as you have suggested. I believe it will still be incomprehensibly small for a human mind. With some effort it should be possible to represent this fact mathematically and build models to deal with situations at that micro level. I think people have done that.


Dear Shri Vaagmi,

I am not saying because we cannot see the distance , that it is zero but what I am saying is, it would seem in theory, that there would always be some space however finely you divide and it would never seem to approach zero.

The way one can explain this is that space is made of units where these units represent some energy and in which the energy is constant. This energy may be something that is built into the structure of space. There is change in energy between units and this manifests as distance and all that one has to do to move in space is to jump from one unit to another by exerting energy. Kindly see the paradox in the next post that poses the same problem

I will come to the second example once we discuss this.
 
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Dear Shri Vaagmi,

I just saw this one and it seems exactly refer to the problem I was talking about:

"In the paradox of
Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise"

Kindly see my previous post for resolving the paradox.
 
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Dear sravna,

You speak about speed. Speed is relative and not absolute. Even leaving that, if you have infinite space you can have infinite speed too. A particle can start from rest and keep on accelerating towards reaching that point at the infinite distance in the infinite space. You have to only wait till infinity for the particle to reach that infinite speed because it will still be accelerating towards infinity. I am going mad. I am just a human being. I have limits.

Some loud thinking by a lay man.Thanks.

Dear Shri Vaagmi,

I think it is not possible to have a meaningful debate if we first do not know what the other understands by the basic notions such as space, time, energy, mass, force, velocity etc . But I am sure that there is no complete or even unanimous understanding of the above even in the scientific community. I think for a discussion at least we need to specify our understanding that too in layman's terms if possible.

I would start with asking what are space and time according to you? to understand what you meant by what you said above.
 
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